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Properties

Label 2-15e2-5.4-c5-0-29
Degree $2$
Conductor $225$
Sign $0.447 + 0.894i$
Analytic cond. $36.0863$
Root an. cond. $6.00719$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 28·4-s − 192i·7-s + 120i·8-s + 148·11-s + 286i·13-s + 384·14-s + 656·16-s − 1.67e3i·17-s − 1.06e3·19-s + 296i·22-s − 2.97e3i·23-s − 572·26-s − 5.37e3i·28-s − 3.41e3·29-s + ⋯
L(s)  = 1  + 0.353i·2-s + 0.875·4-s − 1.48i·7-s + 0.662i·8-s + 0.368·11-s + 0.469i·13-s + 0.523·14-s + 0.640·16-s − 1.40i·17-s − 0.673·19-s + 0.130i·22-s − 1.17i·23-s − 0.165·26-s − 1.29i·28-s − 0.752·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(36.0863\)
Root analytic conductor: \(6.00719\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 225,\ (\ :5/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.228693738\)
\(L(\frac12)\) \(\approx\) \(2.228693738\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 - 2iT - 32T^{2} \)
7 \( 1 + 192iT - 1.68e4T^{2} \)
11 \( 1 - 148T + 1.61e5T^{2} \)
13 \( 1 - 286iT - 3.71e5T^{2} \)
17 \( 1 + 1.67e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.06e3T + 2.47e6T^{2} \)
23 \( 1 + 2.97e3iT - 6.43e6T^{2} \)
29 \( 1 + 3.41e3T + 2.05e7T^{2} \)
31 \( 1 + 2.44e3T + 2.86e7T^{2} \)
37 \( 1 + 182iT - 6.93e7T^{2} \)
41 \( 1 - 9.39e3T + 1.15e8T^{2} \)
43 \( 1 + 1.24e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.20e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.38e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.00e4T + 7.14e8T^{2} \)
61 \( 1 - 3.23e4T + 8.44e8T^{2} \)
67 \( 1 + 6.09e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.26e4T + 1.80e9T^{2} \)
73 \( 1 + 3.87e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.33e4T + 3.07e9T^{2} \)
83 \( 1 + 1.67e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.01e5T + 5.58e9T^{2} \)
97 \( 1 - 1.19e5iT - 8.58e9T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.10405029313746446887448562810, −10.44596667286359341322326001092, −9.261877394176351524991132880577, −7.905920971567429481788373155079, −7.06749468039505220342846903490, −6.42498860151287064232203569582, −4.89077936787222148762140689743, −3.65966655599419577346465504890, −2.13731800534778857489148889745, −0.62298110197074692411003408939, 1.53671309530455615790778619244, 2.54542137291725847544868945281, 3.78157151679034175273047986240, 5.60869914040530507862078919076, 6.23799340076894017166890343514, 7.58428022467062287266018046946, 8.658584359998800304854877577487, 9.654759759568846321968885184156, 10.78708055024353268010157070845, 11.52754875252598540989743484064

Graph of the $Z$-function along the critical line